amvevd: Parametric Dependence Functions of Multivariate Extreme Value...

In evd: Functions for Extreme Value Distributions

Parametric Dependence Functions of Multivariate Extreme Value Models

Description

Calculate the dependence function A for the multivariate logistic and multivariate asymmetric logistic models; plot the estimated function in the trivariate case.

Usage

amvevd(x = rep(1/d,d), dep, asy, model = c("log", "alog"), d = 3, plot =
    FALSE, col = heat.colors(12), blty = 0, grid = if(blty) 150 else 50,
    lower = 1/3, ord = 1:3, lab = as.character(1:3), lcex = 1)

Arguments

x	A vector of length d or a matrix with d columns, in which case the dependence function is evaluated across the rows (ignored if plot is TRUE). The elements/rows of the vector/matrix should be positive and should sum to one, or else they should have a positive sum, in which case the rows are rescaled and a warning is given. A(1/d,\dots,1/d) is returned by default since it is often a useful summary of dependence.
dep	The dependence parameter(s). For the logistic model, should be a single value. For the asymmetric logistic model, should be a vector of length 2^d-d-1, or a single value, in which case the value is used for each of the 2^d-d-1 parameters (see rmvevd).
asy	The asymmetry parameters for the asymmetric logistic model. Should be a list with 2^d-1 vector elements containing the asymmetry parameters for each separate component (see rmvevd and Examples).
model	The specified model; a character string. Must be either "log" (the default) or "alog" (or any unique partial match), for the logistic and asymmetric logistic models respectively. The definition of each model is given in rmvevd.
d	The dimension; an integer greater than or equal to two. The trivariate case d = 3 is the default.
plot	Logical; if TRUE, and the dimension d is three (the default dimension), the dependence function of a trivariate model is plotted. For plotting in the bivariate case, use abvevd. If FALSE (the default), the following arguments are ignored.
col	A list of colours (see image). The first colours in the list represent smaller values, and hence stronger dependence. Each colour represents an equally spaced interval between lower and one.
blty	The border line type, for the border that surrounds the triangular image. By default blty is zero, so no border is plotted. Plotting a border leads to (by default) an increase in grid (and hence computation time), to ensure that the image fits within it.
grid	For plotting, the function is evaluated at grid^2 points.
lower	The minimum value for which colours are plotted. By defualt \code{lower} = 1/3 as this is the theoretical minimum of the dependence function of the trivariate extreme value distribution.
ord	A vector of length three, which should be a permutation of the set \{1,2,3\}. The points (1,0,0), (0,1,0) and (0,0,1) (the vertices of the simplex) are depicted clockwise from the top in the order defined by ord.The argument alters the way in which the function is plotted; it does not change the function definition.
lab	A character vector of length three, in which case the ith margin is labelled using the ith component, or NULL, in which case no labels are given. The actual location of the margins, and hence the labels, is defined by ord.
lcex	A numerical value giving the amount by which the labels should be scaled relative to the default. Ignored if lab is NULL.

Details

Let z = (z_1,\dots,z_d) and w = (w_1,\dots,w_d). Any multivariate extreme value distribution can be written as

G(z) = \exp\left\{- \left\{\sum\nolimits_{j=1}^{d} y_j \right\} A\left(\frac{y_1}{\sum\nolimits_{j=1}^{d} y_j}, \dots, \frac{y_d}{\sum\nolimits_{j=1}^{d} y_j}\right)\right\}

for some function A defined on the simplex S_d = \{w \in R^d_+ : \sum\nolimits_{j=1}^{d} w_j = 1\}, where

y_i = \{1+s_i(z_i-a_i)/b_i\}^{-1/s_i}

for 1+s_i(z_i-a_i)/b_i > 0 and i = 1,\dots,d, and where the (generalized extreme value) marginal parameters are given by (a_i,b_i,s_i), b_i > 0. If s_i = 0 then y_i is defined by continuity.

A is called (by some authors) the dependence function. It follows that A(w) = 1 when w is one of the d vertices of S_d, and that A is a convex function with \max(w_1,\dots,w_d) \leq A(w)\leq 1 for all w in S_d. The lower and upper limits of A are obtained under complete dependence and mutual independence respectively. A does not depend on the marginal parameters.

Value

A numeric vector of values. If plotting, the smallest evaluated function value is returned invisibly.

See Also

amvnonpar, abvevd, rmvevd, image

Examples

amvevd(dep = 0.5, model = "log")
s3pts <- matrix(rexp(30), nrow = 10, ncol = 3)
s3pts <- s3pts/rowSums(s3pts)
amvevd(s3pts, dep = 0.5, model = "log")
## Not run: amvevd(dep = 0.05, model = "log", plot = TRUE, blty = 1)
amvevd(dep = 0.95, model = "log", plot = TRUE, lower = 0.94)

asy <- list(.4, .1, .6, c(.3,.2), c(.1,.1), c(.4,.1), c(.2,.3,.2))
amvevd(s3pts, dep = 0.15, asy = asy, model = "alog")
amvevd(dep = 0.15, asy = asy, model = "al", plot = TRUE, lower = 0.7)

evd documentation built on Sept. 21, 2024, 9:06 a.m.